The tourists looked deflated after skipper Alastair Cook dropped two catches as England searched for early breakthroughs to put pressure on the Australians, who were chasing 201 runs on the fourth day to win the fourth Ashes Test.
The two fielding blunders from the captain were psychological body blows for England and evidence of batsman Kevin Pietersen's comments on Saturday in a television interview that the tourists were "mentally fragile".
Australia clinched the Ashes inside 14 playing days with huge Test wins in Brisbane, Adelaide and Perth, and England appeared incapable of preventing Michael Clarke's team from taking a 4-0 series lead ahead of the fifth and final Test in Sydney later this week.
Cook got his right hand to a snick from Rogers on 19 through slips after wicketkeeper Jonny Bairstow failed to respond, and then the England captain dropped a straightforward two-handed chance from David Warner (22) off Stuart Broad.
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Rogers went on to score his fourth half-century of the series on the ground where he plays for Victoria in the Sheffield Shield competition.
Australia's surge to apparent victory comes after a dramatic turnaround in fortunes on Saturday's third day when England, holding a 51 innings lead, were bowled out for 179, leaving the home side 231 runs to win the Test.
Symbolic of England's malaise, Cook did not bring on his specialist spinner Monty Panesar until 90 minutes into the fourth day and he had little effect on stemming the Australian run flow, with his four overs going for 21 runs.